العمليات في مجموعة الأعداد الكسرية

السنة الثامنة أساسي - تمارين مع الأعداد الكسرية السالبة

تمرين عدد 1: احسب المجاميع التالية
$A = -\frac{7}{4} + \frac{3}{4}$
$B = \frac{5}{3} + \left(-\frac{11}{3}\right)$
$C = -\frac{8}{5} + \frac{2}{5}$
$D = -\frac{9}{7} + \left(-\frac{5}{7}\right)$
$E = \frac{13}{6} + \left(-\frac{7}{6}\right)$
$F = -\frac{15}{8} + \left(-\frac{1}{8}\right)$

الحل التفصيلي

$A = -\frac{7}{4} + \frac{3}{4} = \frac{-7+3}{4} = \frac{-4}{4} = -1$
$B = \frac{5}{3} + \left(-\frac{11}{3}\right) = \frac{5-11}{3} = \frac{-6}{3} = -2$
$C = -\frac{8}{5} + \frac{2}{5} = \frac{-8+2}{5} = \frac{-6}{5} = -\frac{6}{5}$
$D = -\frac{9}{7} + \left(-\frac{5}{7}\right) = \frac{-9-5}{7} = \frac{-14}{7} = -2$
$E = \frac{13}{6} + \left(-\frac{7}{6}\right) = \frac{13-7}{6} = \frac{6}{6} = 1$
$F = -\frac{15}{8} + \left(-\frac{1}{8}\right) = \frac{-15-1}{8} = \frac{-16}{8} = -2$
تمرين عدد 2: احسب الفرق التالية
$A = -\frac{5}{6} - \frac{7}{6}$
$B = \frac{9}{4} - \left(-\frac{3}{4}\right)$
$C = -\frac{11}{5} - \left(-\frac{1}{5}\right)$
$D = -\frac{7}{8} - \frac{9}{8}$
$E = \frac{4}{3} - \left(-\frac{8}{3}\right)$
$F = -\frac{13}{7} - \left(-\frac{6}{7}\right)$

الحل التفصيلي

$A = -\frac{5}{6} - \frac{7}{6} = \frac{-5-7}{6} = \frac{-12}{6} = -2$
$B = \frac{9}{4} - \left(-\frac{3}{4}\right) = \frac{9}{4} + \frac{3}{4} = \frac{12}{4} = 3$
$C = -\frac{11}{5} - \left(-\frac{1}{5}\right) = -\frac{11}{5} + \frac{1}{5} = \frac{-10}{5} = -2$
$D = -\frac{7}{8} - \frac{9}{8} = \frac{-7-9}{8} = \frac{-16}{8} = -2$
$E = \frac{4}{3} - \left(-\frac{8}{3}\right) = \frac{4}{3} + \frac{8}{3} = \frac{12}{3} = 4$
$F = -\frac{13}{7} - \left(-\frac{6}{7}\right) = -\frac{13}{7} + \frac{6}{7} = \frac{-7}{7} = -1$
تمرين عدد 3: احسب المجاميع مع مقامات مختلفة
$A = -\frac{3}{4} + \frac{5}{6}$
$B = \frac{7}{8} + \left(-\frac{3}{4}\right)$
$C = -\frac{5}{6} + \left(-\frac{2}{3}\right)$
$D = -\frac{7}{10} + \frac{3}{5}$
$E = \frac{11}{12} + \left(-\frac{2}{3}\right)$
$F = -\frac{9}{8} + \left(-\frac{5}{6}\right)$

الحل التفصيلي

$A = -\frac{3}{4} + \frac{5}{6} = \frac{-9}{12} + \frac{10}{12} = \frac{1}{12}$
$B = \frac{7}{8} + \left(-\frac{3}{4}\right) = \frac{7}{8} - \frac{6}{8} = \frac{1}{8}$
$C = -\frac{5}{6} + \left(-\frac{2}{3}\right) = \frac{-5}{6} - \frac{4}{6} = \frac{-9}{6} = -\frac{3}{2}$
$D = -\frac{7}{10} + \frac{3}{5} = \frac{-7}{10} + \frac{6}{10} = \frac{-1}{10} = -\frac{1}{10}$
$E = \frac{11}{12} + \left(-\frac{2}{3}\right) = \frac{11}{12} - \frac{8}{12} = \frac{3}{12} = \frac{1}{4}$
$F = -\frac{9}{8} + \left(-\frac{5}{6}\right) = \frac{-27}{24} - \frac{20}{24} = \frac{-47}{24} = -\frac{47}{24}$
تمرين عدد 4: احسب الفرق مع مقامات مختلفة
$A = -\frac{5}{6} - \frac{3}{4}$
$B = \frac{9}{10} - \left(-\frac{2}{5}\right)$
$C = -\frac{7}{12} - \left(-\frac{5}{8}\right)$
$D = -\frac{11}{9} - \frac{5}{6}$
$E = \frac{13}{15} - \left(-\frac{4}{5}\right)$
$F = -\frac{17}{20} - \left(-\frac{3}{4}\right)$

الحل التفصيلي

$A = -\frac{5}{6} - \frac{3}{4} = \frac{-10}{12} - \frac{9}{12} = \frac{-19}{12} = -\frac{19}{12}$
$B = \frac{9}{10} - \left(-\frac{2}{5}\right) = \frac{9}{10} + \frac{4}{10} = \frac{13}{10}$
$C = -\frac{7}{12} - \left(-\frac{5}{8}\right) = -\frac{14}{24} + \frac{15}{24} = \frac{1}{24}$
$D = -\frac{11}{9} - \frac{5}{6} = \frac{-22}{18} - \frac{15}{18} = \frac{-37}{18} = -\frac{37}{18}$
$E = \frac{13}{15} - \left(-\frac{4}{5}\right) = \frac{13}{15} + \frac{12}{15} = \frac{25}{15} = \frac{5}{3}$
$F = -\frac{17}{20} - \left(-\frac{3}{4}\right) = -\frac{17}{20} + \frac{15}{20} = \frac{-2}{20} = -\frac{1}{10}$
تمرين عدد 5: بسّط ثم احسب العبارات التالية
$A = -\frac{5}{3} + \frac{1127}{1126} - 1 - \frac{1127}{1126}$
$B = -\frac{13}{25} + \frac{7}{19} - \frac{7}{19} - \frac{2}{5}$
$C = -2 + \frac{11}{4} + 3 - \frac{33}{12}$
$D = -\frac{9}{2} - \frac{7}{6} + \frac{14}{12} + \frac{15}{2}$

الحل التفصيلي

$A = -\frac{5}{3} + \frac{1127}{1126} - 1 - \frac{1127}{1126} = -\frac{5}{3} - 1 = -\frac{5+3}{3} = -\frac{8}{3}$
$B = -\frac{13}{25} + \frac{7}{19} - \frac{7}{19} - \frac{2}{5} = -\frac{13}{25} - \frac{2}{5} = -\frac{13}{25} - \frac{10}{25} = -\frac{23}{25}$
$C = -2 + \frac{11}{4} + 3 - \frac{33}{12} = 1 + \frac{11}{4} - \frac{11}{4} = 1$
$D = -\frac{9}{2} - \frac{7}{6} + \frac{14}{12} + \frac{15}{2} = \frac{6}{2} + \frac{14-14}{12} = 3$
تمرين عدد 6: ابحث عن العدد الكسري x في كل حالة من الحالات التالية
$-\frac{7}{4} + x = 1$
$-\frac{9}{2} + x = -5$
$x + \left(-\frac{5}{13}\right) = -1$
$x + \frac{3}{7} = -\frac{11}{14}$
$-\frac{17}{30} + x = \frac{13}{40}$
$x + \left(-\frac{19}{28}\right) = -\frac{15}{21}$

الحل التفصيلي

$-\frac{7}{4} + x = 1 \Rightarrow x = 1 + \frac{7}{4} = \frac{4+7}{4} = \frac{11}{4}$
$-\frac{9}{2} + x = -5 \Rightarrow x = -5 + \frac{9}{2} = \frac{-10+9}{2} = -\frac{1}{2}$
$x + \left(-\frac{5}{13}\right) = -1 \Rightarrow x = -1 + \frac{5}{13} = \frac{-13+5}{13} = -\frac{8}{13}$
$x + \frac{3}{7} = -\frac{11}{14} \Rightarrow x = -\frac{11}{14} - \frac{3}{7} = \frac{-11-6}{14} = -\frac{17}{14}$
$-\frac{17}{30} + x = \frac{13}{40} \Rightarrow x = \frac{13}{40} + \frac{17}{30} = \frac{39+68}{120} = \frac{107}{120}$
$x + \left(-\frac{19}{28}\right) = -\frac{15}{21} \Rightarrow x = -\frac{5}{7} + \frac{19}{28} = \frac{-20+19}{28} = -\frac{1}{28}$
تمرين عدد 7: عمليات متنوعة مع أعداد كسرية سالبة
$A = -\frac{5}{6} + \frac{7}{9} - \frac{2}{3}$
$B = \frac{11}{8} - \left(-\frac{3}{4}\right) + \left(-\frac{9}{8}\right)$
$C = -\frac{7}{5} - \frac{3}{10} + \frac{9}{10}$
$D = -\frac{13}{12} + \left(-\frac{5}{6}\right) - \left(-\frac{7}{4}\right)$

الحل التفصيلي

$A = -\frac{5}{6} + \frac{7}{9} - \frac{2}{3} = \frac{-15}{18} + \frac{14}{18} - \frac{12}{18} = \frac{-13}{18}$
$B = \frac{11}{8} - \left(-\frac{3}{4}\right) + \left(-\frac{9}{8}\right) = \frac{11}{8} + \frac{6}{8} - \frac{9}{8} = \frac{8}{8} = 1$
$C = -\frac{7}{5} - \frac{3}{10} + \frac{9}{10} = \frac{-14}{10} - \frac{3}{10} + \frac{9}{10} = \frac{-8}{10} = -\frac{4}{5}$
$D = -\frac{13}{12} + \left(-\frac{5}{6}\right) - \left(-\frac{7}{4}\right) = -\frac{13}{12} - \frac{10}{12} + \frac{21}{12} = \frac{-2}{12} = -\frac{1}{6}$
تمرين عدد 8: حل المعادلات مع أعداد كسرية سالبة
$-\frac{5}{3} + x = -\frac{7}{6}$
$x - \left(-\frac{8}{5}\right) = \frac{3}{10}$
$-\frac{11}{4} - x = -\frac{9}{8}$
$x + \frac{7}{12} = -\frac{5}{6}$

الحل التفصيلي

$-\frac{5}{3} + x = -\frac{7}{6} \Rightarrow x = -\frac{7}{6} + \frac{5}{3} = \frac{-7+10}{6} = \frac{3}{6} = \frac{1}{2}$
$x - \left(-\frac{8}{5}\right) = \frac{3}{10} \Rightarrow x = \frac{3}{10} - \frac{8}{5} = \frac{3-16}{10} = -\frac{13}{10}$
$-\frac{11}{4} - x = -\frac{9}{8} \Rightarrow x = -\frac{11}{4} + \frac{9}{8} = \frac{-22+9}{8} = -\frac{13}{8}$
$x + \frac{7}{12} = -\frac{5}{6} \Rightarrow x = -\frac{5}{6} - \frac{7}{12} = \frac{-10-7}{12} = -\frac{17}{12}$